144 KINETICS OF A RIGID BODY. [264. 



radii vectores of the momental ellipsoid at the point, gives at 

 once a number of propositions about these moments. It is 

 only necessary to interpret properly the geometrical properties 

 of the ellipsoid. Thus, it is known that the sum of the squares 

 of the reciprocals of any three rectangular semi-diameters of an 

 ellipsoid is constant. It follows that the sum of the three 

 moments of inertia for any three rectangular lines passing 

 through the same point has a constant value. 



In general, the three principal moments of inertia f v 7 2 , 7 3 

 at a point O are different. If, however, two of them are equal, 

 say / 2 = / 3 , the momental ellipsoid becomes an ellipsoid of 

 revolution about the third, f lt as axis ; and it follows that the 

 moments of inertia for all lines through O lying in the plane 

 of the two equal axes are equal. 



If 1^ = 1^ = 1^ the ellipsoid becomes a sphere, and the mo- 

 ments of inertia are the same for all lines passing through O. 



264. If the equation of the momental ellipsoid at a point O 

 be of the form Ax* + B}P+Cz* 2Dyz = J/e 4 ; i.e. if the two con- 

 ditions 



E = ^mzx = o, F = ^mxy o 



be fulfilled, the axis of x coincides with one of the three axes 

 of the ellipsoid, the surface being symmetrical with respect to 

 the ^-plane. Hence, if the conditions E = o, F = o are satisfied, 

 the axis of K is a principal axis at the origin. The converse is 

 evidently also true ; i.e. if a line is a principal axis at one of 

 its points, then, taking this point as origin and the line as axis 

 of x t the conditions ^m^ = o, ^mxy = o must be satisfied. 



It is easy to see that if a line be a principal axis at one of its 

 points, say O, it wjll in general not be a principal axis at any 

 other one of its points. For, taking the line as axis of x and 



as origin, we have ^mzx=o t ^mxy = o. If now for a point 



1 on this line at the distance a from O the line is likewise 

 a principal axis, the conditions 



